An introduction to the theory of random graphs

This thesis provides an introduction to the fundamentals of random graph theory. The study starts introduces the two fundamental building blocks of random graph theory, namely discrete probability and graph theory. The study starts by introducing relevant concepts probability commonly used in random graph theory- these include concentration inequalities such as Chebyshev's inequality and Chernoff's inequality. Moreover we proceed by introducing central concepts in graph theory, which will underpin the later discussion. In particular we provide results such as Mycielski's construction of a family of triangle-free graphs with high chromatic number and results in Ramsey theory. Next we introduce the concept of a random graph and present two of the most famous proofs in graph theory using the theory random graphs. These include the proof of the fact that there are graphs with arbitrarily high girth and chromatic number, and a bound on the Ramsey number $R$($k$; $k$). Finally we conclude by introducing the notion of a threshold function for a monotone graph property and we present proofs for the threhold functions of certain properties

Independent sets in Line of Sight networks

Line of Sight (LoS) networks provide a model of wireless communication which incor-porates visibility constraints. Vertices of such networks can be embedded onto the cube{(x1,x2,...,xd):xi∈{1,...,n},1≤i≤d}so that two vertices are adjacent if and onlyif their images lay on a line parallel to one of the cube edges and their distance is lessthan a given range parameterω. In this paper we study large independent sets in LoSnetworks.Weprovethatthecomputationalproblemoffindingamaximumindependentset can be solved optimally in polynomial time for one dimensional LoS networks.However, ford≥2, the (decision version of) the problem becomes NP-complete for anyfixedω≥3. In addition, we show that the problem is APX-hard whenω=nford≥3.On the positive side, we show that LoS networks generalize chordal graphs. This impliesthat there exists a simpled-approximation algorithm for the maximum independent setproblem in LoS networks. Finally, we describe a polynomial time approximation schemefor the maximum independent set problem in LoS networks for the case whenωis aconstantandpresentanimprovedheuristicalgorithmfortheprobleminthecaseω=

Dynamic Programming Optimization in Line of Sight Networks

Line of Sight (LoS) networks were designed to model wireless communication in settings which may contain obstacles. For fixed positive integer d, and positive integer ω, a graph is a (d-dimensional) LoS network with range parameter ω if it can be embedded in a finite cube of the d-dimensional integer grid so that each pair of vertices in V are adjacent if and only if their embedding coordinates differ only in one position and such difference is less than ω. In this paper we investigate a dynamic programming (DP) approach which can be used to obtain efficient algorithmic solutions for various combinatorial problems in LoS networks. In particular DP solves the Maximum Independent Set (MIS) problem in LoS networks optimally, for any ω, on narrow LoS networks (i.e. networks which can be embedded in a region, for some fixed k independent of n). In the unrestricted case it has been shown that the (decision version of the) problem is NP-hard when , for fixed . We describe how DP can be used as a building block in the design of good approximation algorithms in this case. In particular we present a semi-online polynomial-time approximation scheme for the MIS problem in narrow d-dimensional LoS networks, as well as a polynomial-time 2-approximation algorithm and a fast polynomial time approximation scheme for the MIS problem in arbitrary d-dimensional LoS networks. Finally we comment on how the approach can be adapted to prove similar results for a number of important optimization problems in LoS networks

Dynamic Programming Optimization in Line of Sight Networks

Line of Sight (LoS) networks were designed to model wireless communication in settings which may contain obstacles restricting node visibility. For fixed positive integer $d$, and positive integer $\omega$, a graph $G=(V,E)$ is a ($d$-dimensional) LoS network with range parameter $\omega$ if it can be embedded in a cube of side size $n$ of the $d$-dimensional integer grid so that each pair of vertices in $V$ are adjacent if and only if their embedding coordinates differ only in one position and such difference is less than $\omega$. In this paper we investigate a dynamic programming (DP) approach which can be used to obtain efficient algorithmic solutions for various combinatorial problems in LoS networks. In particular DP solves the Maximum Independent Set (MIS) problem in LoS networks optimally for any $\omega$ on {\em narrow} LoS networks (i.e. networks which can be embedded in a $n \times k \times k \ldots \times k$ region, for some fixed $k$ independent of $n$). In the unrestricted case it has been shown that the MIS problem is NP-hard when $\omega > 2$ (the hardness proof goes through for any $\omega=O(n^{1-\delta})$, for fixed $0<\delta<1$). We describe how DP can be used as a building block in the design of good approximation algorithms. In particular we present a 2-approximation algorithm and a fast polynomial time approximation scheme for the MIS problem in arbitrary $d$-dimensional LoS networks. Finally we comment on how the approach can be adapted to solve a number of important optimization problems in LoS networks